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A linear-quadratic optimal-control problem has linear state dynamics and running and terminal costs defined by quadratic forms. Its value function is quadratic in the state, the optimal feedback is linear, and its coefficient obeys a Riccati equation.

A smooth manifold carrying a nondegenerate bilinear form on each tangent space, with one negative and all remaining positive eigenvalues, or the overall reversed-sign convention. Its nonzero null vectors have vanishing metric quadratic form.

For a static smooth ideal magnetohydrodynamics equilibrium with conservative external forces and surface terms eliminated by the boundary conditions, the linear displacement equation is $\rho \ddot{\mathbf{\xi }}=\mathcal{F}\mathbf{\xi }$, where the force operator is a self-adjoint operator in the appropriate integral pairing. Its restoring quadratic form $\delta W=-\frac{1}{2}\int {\mathbf{\xi }}^{\ast }\cdot \mathcal{F}\mathbf{\xi }dV$ is real. Multiplying the normal mode equation by the complex conjugate of the displacement and integrating gives ${\omega }^2\int \rho |\mathbf{\xi }{|}^{2}dV=2\delta W$. Hence squared frequencies are real, and a negative restoring form gives an unstable mode through the variational Rayleigh quotient, subject to the operator domain and spectral hypotheses. Background fluid flow adds terms to this argument; the static hypothesis is essential.

The mass matrix is the Gram matrix of a finite element basis in the L2 inner product. Its coefficient quadratic form is the squared L2 norm of the represented trial function, so it is a positive-definite matrix for a linearly independent basis.

Two symmetric matrices are congruent when $C=P^{T}AP$ for an invertible matrix $P$. This is the transformation of a quadratic form under an invertible linear change of coordinates. It differs from matrix similarity, which uses $P^{-1}AP$. Completing the square diagonalizes a real quadratic form by matrix congruence.

For a real quadratic form on an $n$-dimensional vector space, with rank $r$ and signature $s$ defined as positive minus negative index, the maximum dimension of a totally isotropic subspace is $n-(r+|s|)/2$. Quotienting by the radical of a bilinear form leaves a nondegenerate form of indices $p,q$. Projection of an isotropic subspace to each sign-coordinate space is injective, bounding its dimension by $\min (p,q)$. The radical and vectors pairing one positive with one negative unit coordinate attain the bound $n-r+\min (p,q)$.

A nondivergence-form elliptic operator applies its principal coefficients directly to the Hessian matrix: $Lu=a^{ij}{D}_{ij}u+b^i{D}_{i}u+cu$. Only the symmetric part of $a$ contributes to the second-derivative term. Uniform ellipticity bounds its quadratic form below by $\lambda |\xi {|}^2$ with a constant $\lambda >0$ independent of position.

An odd trial state is orthogonal to the even oscillator ground state. Its Rayleigh quotient therefore bounds the first excited energy from above, unlike an arbitrary trial state. The kinetic quadratic form permits continuous compactly supported trials with derivative jumps.